A Fredholm operator is a type of linear operator that arises in functional analysis, particularly in the study of Hilbert spaces and Banach spaces. It is characterized by having a finite-dimensional kernel (the set of vectors that it maps to zero) and a closed range. This means that the operator can be inverted on its range, leading to important implications in solving equations. Fredholm operators are classified into three types based on their index, which is the difference between the dimension of the kernel and the dimension of the cokernel (the quotient of the codomain by the range). This classification helps in understanding the solvability of linear equations and the stability of solutions in various mathematical contexts.